CGAL users discussion list

[Tapadi <>]

Hi,

Discretize the polygon means dividing it into small pieces like triangles,
such as Andreas proposes.

Once you cannot compute the exact integral of your function over one
triangle, you'll compute an approximation of it. This is "discretizing the
density function". Then, you'll sum all approximated integrals to make up a
global approximated integral of your density function over the polygon.

An example of approximated integral over a triangle is:
[A]   "value of the function at center of the triangle" x "area of the
triangle"

One can easily deduce that the smaller are the triangles, the more accurate
the final result will be. Hence, you can divide your triangles again and
again in order to reduce the error of the approximation.

What Andreas proposes is a better way to divide the triangles. The
approximation [A] means that you consider your function to be constant over
the triangle. To check "how much" this is true, you can compare values of
the function at the three triangle's vertices. If they are very different,
then your function isn't constant at all here. Divide the triangle. If they
are around equal, then your function has good chances to be near constant in
the triangle. Don't divide the triangle.

There are many ways to discretize a domain and a function in order to
compute an approximated integral. Just try this simple one, ask again if
something was not clear in my explanation. And check out for the
normalization of your density function: if you sum up many values of it, you
may end with an overflowed result.

Best regards,
Hugo Loi
PhD sudent at Inria - Maverick Team




--
View this message in context: 
http://cgal-discuss.949826.n4.nabble.com/Area-of-convex-polygons-with-non-constant-density-tp4657579p4657586.html
Sent from the cgal-discuss mailing list archive at Nabble.com.